A Perturbation-Based Method for Extracting Elastic Properties during Spherical Indentation of an Elastic Film/Substrate
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A Perturbation-Based Method for Extracting Elastic Properties during Spherical Indentation of an Elastic Film/Substrate Bilayer Jae Hun Kim1 , Andrew Gouldstone2 and Chad S. Korach3 1 Department of Materials Science and Engineering, Stony Brook University, Stony Brook, NY 11794, U.S.A., 2Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02199, U.S.A., 3Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794, U.S.A. ABSTRACT Accurate mechanical property measurement of films on substrates by instrumented indentation requires a solution describing the effective modulus of the film/substrate system. Here, a firstorder elastic perturbation solution for spherical punch indentation on a film/substrate system is presented. Finite element method (FEM) simulations were conducted for comparison with the analytic solution. FEM results indicate that the new solution is valid for a practical range of modulus mismatch, especially for a stiff film on a compliant substrate. INTRODUCTION Measurement of mechanical properties of films deposited on substrates has long been an issue in thin- and thick-film technology. Although indentation is extensively used due to its relative experimental simplicity, analysis is complicated by the inevitable substrate effect. Rules of practice exist that state film properties may be isolated if contact dimensions are small compared to film thickness, but such simplifications are not useful for layers including microstructural size effects, or ultra-thin films. Thus, analyses that consider the relation between film and substrate properties are necessary.
Figure 1. Spherical indentation of an elastic film-substrate bilayer. The force-depth response will be governed by the elastic properties of both constituents. Consider the indentation of an ideally elastic film/substrate system that is mechanically bonded (Fig. 1). A proper description of the effective modulus µ eff of the film/substrate system, in the context of film and substrate moduli ( µ f and µ s ) is essential to the accurate extraction of the film properties. Since there is no exact solution yet for this problem, a number of approximate models have been proposed to be fitted with data from experiments, and FEM simulations, or analytic solutions for several tip geometries. Those models have shown good yet limited applicability. Most of these approaches were based on the following simple formula:
µeff = µ s + (µ f − µ s )φ (a / t , µ f / µ s )
(1)
where a is contact radius, t is the thickness of the film and φ is a weight function of contact size, tip geometry and modulus mismatch; φ approaches 1 when a / t → 0 and 0 when a / t → ∞ . Numerous forms of Eq (1) have been proposed for sharp and flat-ended cylindrical geometry. For brevity, they are grouped and listed here. • Doerner and Nix[1] first introduced an empirical equation of an effective modulus expressed as an exponential form and fit it with experimental data obtained with a sharp tip. King[2] performed
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